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Bill Trok
Bill Trok
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The following series arose in some work related to Hilbert Functions of ideals of points

$$\sum_{k = 0}^m (-1)^k {2m+2 \choose k}[2m(m+1)-k(2m+1)]^{2m-1}.$$

Experimentally, this series is always negative for $m > 0$$m > 1$ and decreases incredibly quickly. We only need that this is never $0$. We have tried rational roots, taking first differences, but have been unable to make much progress. Any help or suggestions are appreciated.

The following series arose in some work related to Hilbert Functions of ideals of points

$$\sum_{k = 0}^m (-1)^k {2m+2 \choose k}[2m(m+1)-k(2m+1)]^{2m-1}.$$

Experimentally, this series is always negative for $m > 0$ and decreases incredibly quickly. We only need that this is never $0$. We have tried rational roots, taking first differences, but have been unable to make much progress. Any help or suggestions are appreciated.

The following series arose in some work related to Hilbert Functions of ideals of points

$$\sum_{k = 0}^m (-1)^k {2m+2 \choose k}[2m(m+1)-k(2m+1)]^{2m-1}.$$

Experimentally, this series is always negative for $m > 1$ and decreases incredibly quickly. We only need that this is never $0$. We have tried rational roots, taking first differences, but have been unable to make much progress. Any help or suggestions are appreciated.

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Bill Trok
Bill Trok
  • 201
  • 1
  • 5

Showing an alternating integer series is never $0$

The following series arose in some work related to Hilbert Functions of ideals of points

$$\sum_{k = 0}^m (-1)^k {2m+2 \choose k}[2m(m+1)-k(2m+1)]^{2m-1}.$$

Experimentally, this series is always negative for $m > 0$ and decreases incredibly quickly. We only need that this is never $0$. We have tried rational roots, taking first differences, but have been unable to make much progress. Any help or suggestions are appreciated.